In a survey of a group of men, the heights in the 20-29 age group were normally distributed, with a mean of 69.5inches and a standard deviation of 3.0inches. A study participant is randomly selected. Complete parts (a) through (d) below.

a. Find the probability that a study participant has a height that is less than 66 inches.
b. Find the probability that a study participant has a height that is between 66 and 71 inches.
c. Find the probability that a study participant has a height that is more than 71 inches.
d. Identify any unusual events. Explain your reasoning.

a)

And we can find this probability using the normal standard table:

b)

And we can find this probability with thi difference:

And in order to find these probabilities we can use tables for the normal standard distribution, excel or a calculator.  

c)

And we can find this probability using the complement rule and the normal standard table:

d)

If a value is lower than 63.5 or higher than 75.5 we can consider it as unusual

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Part a

Let X the random variable that represent the heights of a population, and for this case we know the distribution for X is given by:

Where and

We are interested on this probability

And the best way to solve this problem is using the normal standard distribution and the z score given by:

If we apply this formula to our probability we got this:

And we can find this probability using the normal standard table:

Part b

And we can find this probability with thi difference:

And in order to find these probabilities we can use tables for the normal standard distribution, excel or a calculator.  

Part c

And we can find this probability using the complement rule and the normal standard table:

Part d

For this case we can consider as unusual events values above/ below two deviations within the mean:

If a value is lower than 63.5 or higher than 75.5 we can consider it as unusual